Abstract: Manifolds with special geometries can be described by their holonomy representation. The  irreducible  holonomy representations of (simply-connected) Riemannian and pseudo-Riemannian manifolds are well known and geometric implications are intensively studied. In the pseudo-Riemannian case a new type of holonomy representations appears, the weakly irreducible but non-irreducible ones, which are - contrary to the irreducible case - not completely classified and geometrically less understood. In the same way as the holonomy of a metric, the holonomy of a conformal structure is defined (using the unique normal conformal Cartan connection).
In the 3. period of the project we will focus on the construction of special metrics (resp. conformal classes) with prescribed holonomy and prescribed causality or symmetry properties.

     Special Research Semester at the International Erwin Schrödinger Institute for Mathematical Physics (ESI) in Vienna, 01.09.2005 -  31.12.2005.
     Organizers: Dmitri Alekseevski (Hull, GB), Helga Baum (HU Berlin) , Jerzy Konderak (Bari, Italy)

Abstract: The aim of the program is to bring together highly qualified researchers working on different aspects of the geometry of manifolds with indefinite metrics. Whereas in Riemannian geometry  in the last 30 years an essential progress was made in the investigation and classification of different classes of Riemannian manifolds (e.g. manifolds with an additional geometric structures, manifolds with conditions on curvature, homogeneous Riemannian spaces), similar results for pseudo-Riemannian manifolds are rare and many problems are still open. For a long time  the main source of problems in pseudo-Riemannian geometry was General Relativity which deals with 4-dimensional Lorentzian manifolds (space-times). However, the developments in Theoretical Physics (Supergravity, String theory) require a deeper understanding of the geometric structure of higher dimensional manifolds with indefinite metrics of different signature. Sometimes, one can use special Ansatzes or "Wickrotations" to transform the problems of pseudo-Riemannian geometry  into  Riemannian ones. But in many aspects pseudo-Riemannian geometry essentially differs from Riemannian one  and many specific questions  appear . In the last time different groups started to focus their attention to geometric problems arising by indefinite metrics. During the program we want to discuss the results of different groups, we want to develop the field by joint research during the program and we want to establish a closer cooperation between the groups.
Main Topics are:

• Holonomy representations of pseudo-Riemannian manifolds
• Geometry of pseudo-Riemannian manifolds with special holonomy
• Transformation groups and infinitesimal symmetries of pseudo-Riemannian manifolds
• Spin geometry of pseudo-Riemannian manifolds, Dirac- and twistor equations
• Geometry of submanifolds
• Geodesics
• Supergravity